## Mayer-Vietoris homotopy groups sequence of a pull-back of a fibration

This must be an elementary question: could somebody tell me a reference for the Mayer-Vietoris homotopy groups sequence of a pull-back of a fibration?

I'm working in the category of pointed simplicial sets. So I've a pull-back of a (Kan) fibration of pointed simplicial sets, and I've read that in this situation you have an associated Mayer-Vietoris sequence relating the homotopy groups of the simplicial sets of the pull-back that looks like the classical Mayer-Vietoris sequence for the singular homology of a pair of open sets covering a topological space.

I've been searching in May's "Simplicial objects in Algebraic Topology" and Goerss-Jardine's "Simplicial Homotopy Theory", but I couldn't find it.

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I don't know of a reference, but here is a quick argument. Suppose we want to compute the homotopy pullback P = X ×hZ Y of two maps f : X → Z and g : Y → Z of pointed simplicial sets. Assume for convenience that everything is fibrant. There is a fibration ZΔ[1] → Z∂Δ[1] = Z × Z with fiber ΩZ. Now P is the pullback of the diagram X × Y → Z × Z ← ZΔ[1]. In particular, P → X × Y is also a fibration with fiber ΩZ, and the Mayer-Vietoris sequence follows from the long exact sequence of homotopy groups of this fibration.

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 Thank you. I was stuck with this point. – Agusti Roig Oct 30 2009 at 10:18 I would add that this is Eckmann-Hilton dual to the argument that gets the usual Mayer-Vietoris sequence from the long exact sequence of a cofibration. – Eric Wofsey Oct 30 2009 at 14:15

A low level Mayer-Vietoris sequence for a pull-back of a fibration of groupoids is in (R. BROWN, P.R. HEATH and H. KAMPS), Groupoids and the Mayer-Vietoris sequence'', {\em J. Pure Appl. Alg.} 30 (1983)

and you will also find a version for coverings of groupoids in Topology and Groupoids', R. Brown (available on amazon.com).

I've set as an exercise in my new coauthored book Nonabelian algebraic topology' (see my web pages) to get a Mayer-Vietoris sequence for a pullback of a fibration of crossed complexes.

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 Thank you: is good to have some references. – Agusti Roig Jul 23 2010 at 17:22